1. Field of Invention
The present invention relates to a motor drive technology, and more particularly to a motor drive technology of a space vector-based current controlled PWM system.
2. Description of Related Arts
Permanent magnet AC motors (PMACMs) have been widely adopted for high-performance servo applications, because of their desirable features: high efficiency, hight torque to inertia ratio, lower maintenance cost, and compact structure when compared to induction and brush DC motors. The use of permanent magnets to generate substantial air gap magnetic flux without excitation makes it possible to design PMACMs with unsurpassed efficiency characteristics. Such efficiency advantages are becoming increasingly valuable in many applications of the world. Since all of the PMACMs are synchronous machines, an average torque can be produced only when the excitation is precisely synchronized with the rotor speed and instantaneous position. The most direct and powerful means of ensuring the synchronization is to continuously measure the rotor's absolute angular position with mounted position sensors, such as Hall-effect sensors, so that the excitation can be switched among the PMACM phases in exact synchronism.
One simple method for achieving the synchronization is using a six-step voltage inverter. The basic operation of the six-step voltage inverter can be understood by considering the inverter as six ideal switches. The line-to-line voltages and line-to-neutral voltages then have the waveform shown in FIG. 1. The line-to-line voltage contains an rms fundamental component of
                              V                      l            ⁢                                                  ⁢                          l              ⁡                              (                                  rms                  ⁢                                                                          ⁢                  fund                                )                                                    =                                            6                        π                    ⁢                      V                          c              ⁢                                                          ⁢              c                                                          (        1        )            
Please refer to J. Holtz, “Pulsewidth modulation—A survey,” IEEE Trans. Ind. Electron., vol. 39, no. 5, pp. 410–420, December 1992. The pulse-width-modulation (PWM) inverter maintains a nearly constant DC link voltage but combines both voltage control and frequency control within the inverter itself. The power switches in the inverter are switched at a high-frequency thus operating, in effect, as choppers. In general, modulation techniques fall into two classes: those that operate at a fixed switching ratio to the fundamental switching frequency and those in which the switching ratio is continuously changing to synthesize a more nearly sinusoidal motor current (called sinusoidal PWM). In the first class, block modulation is the simplest type of modulation and is closest to simple six-step operation. Instead of varying the amplitude of the motor voltage waveform by variation of the DC link voltage, it is varied by switching one or two of the inverter switches at a fixed switching ratio to suit the speed. A simple form of block modulation is shown in FIG. 2, where the chopping is limited to the middle 60 electrical degrees of each device conduction period, resulting in minimum switching duty on the semiconductor switches. In spite of the similarities between block modulation and the six-step mode, the torque pulsations at low speed are much less severe than for the six-step inverter. However, the harmonics of a six-step inverter are also present with block modulation, but there are higher harmonics associated with the chopping frequency of block modulation mode. Hence, the motor losses and noise are significant compared to more elegant modulation algorithms. FIG. 3 shows the phase voltage and current waveforms. Even though the switches TA+ and TA− are in their on state for 180 electrical degrees, due to the lagging power factor of the load, their actual conduction intervals are smaller than 180 electrical degrees.
The second class is the sinusoidal PWM, which is used to synthesize the motor currents as near to sinusoidal waveforms as possible. The lower voltage harmonics can be greatly attenuated, leaving typically only two or four harmonics of substantial amplitude close to the chopping or carrier frequency. With compared to the six-step operation, the motor can rotate much more smoothly at low speed, and the torque pulsations are virtually eliminated and the extra motor losses caused by the inverter are substantially reduced with sinusoidal PWM operation. However, to counterbalance these advantages, the sinusoidal PWM inverter control is complex, and the chopping frequency is high, which causes higher switching losses than the six-step operation. In order to approximate a sine wave, a high-frequency triangular wave is compared with a fundamental frequency sine wave as shown in FIG. 4.
Current control technique plays the most important role in current-controlled PWM inverters, which are widely applied in high-performance motor drives. Various techniques for current controller have been described in the following papers [1]–[7]:    [1] M. Lajoie-Mazenc, C. Villanueva, and J. Hector, “Study and implementation of hysteresis controlled inverter on a permanent magnet synchronous machine,” IEEE Trans. Ind. Applicat., vol. IA-21, no. 2, pp. 408–413, March/April 1985.    [2] D. M. Brod and D. W. Novotny, “Current control of VSI-PWM inverters,” IEEE Trans. Ind. Applicat., vol. IA-21, no. 3, pp. 562–570, May/June 1985.    [3] T. M. Rowan and R. J. Kerkman, “A new synchronous current regulator and an analysis of current-regulated PWM inverters,” IEEE Trans. Ind. Applicat., vol. IA-22, no. 4, pp. 678–690, July/August 1986.    [4] M. P. Kazmierkowski, M. A. Dzieniakowski, and W. Sulkowski, “Novel space vector based current controllers for PWM-inverters,” IEEE Trans. Power Electron., vol. 6, no. 1, pp. 158–166, January 1991.    [5] C. T. Pan and T. Y. Chang, “An improved hysteresis current controller for reducing switching frequency,” IEEE Trans. Power Electron., vol. 9, no. 1, pp. 97–104, 1994.    [6] L. Malesani and P. Tenti, “A novel hysteresis control method for current-controlled voltage-source PWM inverters with constant modulation frequency,” IEEE Trans. Ind. Applicat., vol. 26, no. 1, pp. 88–92, January/February 1990.    [7] S. Buso, S. Fasolo, L. Malesani, and P. Mattavelli, “A dead-beat adaptive hysteresis current control,” IEEE Trans. Ind. Applicat., vol. 36, no. 4, pp. 1174–1180, July/August 2000.
However, among these techniques, the hysteresis current controller (HCC) is a rather popular one because of its easy implementation, fast dynamic response, maximum current limit, and insensitivity to load parameter variations. Nevertheless, depending on load conditions, switching frequency may vary widely during the fundamental period, resulting in irregular inverter operation. This is mainly due to the interference between the commutations of the three phases, since each phase current not only depends on the corresponding phase voltage but is also affected by the voltages of the other two phases. Therefore, the actual current waveform is not only determined by the hysteresis control, but depends on operating conditions. The current slope may vary widely and the current peaks may appreciably exceed the limits of the hysteresis band. The inverter frequency may become much higher than is needed to meet the ripple and noise requirements, and the inverter switches must be rated accordingly. Moreover, high frequency and current peaking increase power loss and may affect system reliability. Some hysteresis current-control techniques applying the zero vector to reduce the number of switchings were reported recently [4]–[5]. Another approach is proposed to minimize the effects of interference between phases while maintaining all the advantages of the hysteresis methods. Due to reduced interference, phase-locked loop (PLL) control of the band amplitude is allowed, giving a constant switching frequency within the period [6]–[7]. However, the control algorithm is more complex and the main advantage of the HCC, i.e. the simplicity, is lost.
On the other hand, the space-vector-modulation (SVM) technique has two excellent features such that its maximum output voltage is 15.4% greater and the number of switchings is about 30% less at the same carrier frequency than the one obtained by the sinusoidal PWM method as described in the following papers and patents [8]–[12].    [8] K. Zhou and D. Wang, “Relationship between space-vector modulation and three-phase carrier-based PWM: A comprehensive analysis,” IEEE Trans. Ind. Electron., vol. 49, no. 1, pp. 186–196, February 2002.    [9] V. Blasko, “Analysis of a hybrid PWM based on modified space-vector and triangle-comparison method,” IEEE Trans. Ind. Applicat., vol. 33, pp. 756–764, May/June 1997.    [10] X. Xu and D. Deng, “Three phase inverter circuit with improved transition from SVPWM to six step operation,” US patent, U.S. Pat. No. 5,552,977, Ford Motor Company, Sep. 3 1996.    [11] V. Blasko, “Hybrid pulse width modulation method and apparatus,” US patent, U.S. Pat. No. 5,706,186, Allen-Bradley Company, Jan. 6 1998.    [12] B. H. Kwon, T. W. Kim, and J. H. Youm, “A novel SVM-based hysteresis current controller,” IEEE Trans. Power Electron., vol. 13, no. 2, pp. 297–307, March 1998.
The SVM technique confines space vectors to be applied according to the region where the output voltage vector is located. However, to obtain the zero-output-current error, the SVM technique requires a measurement of the counter emf vector which is not practical. The HCC can be utilized to make the output-current vector track the command vector with almost negligible response time and insensitivity to line voltage and load parameter variations. However, the HCC generates other vectors except space vectors required according to the region in the SVM technique. If the zero vector is applied to reduce the magnitude of the output-current vector, the line current is decreased with slow slope and the switching frequency is decreased. A SVM-based HCC utilizing all features of the HCC and SVM technique have been developed in [12].
In order to control three phase currents of the motor, an effective method is to measure those directly by three low value resistors or Hall-Effect current sensors. However, this approach is not economical. The number of sensors of the three-phase motor drive can be reduced to two if the motor windings are star connected. However, this method introduces errors in the estimation of the third phase current because of the discrepancies in the gain constants and the DC offset of the other two current sensors. An alternative way is to reconstruct three phase currents based on the measured dc-link current and PWM signals as described in the following papers and patents [13]–[21].    [13] P. P. Acarnley, “Observability criteria for winding currents in three-phase brushless DC drives,” IEEE Trans. Power Electron., vol. 8, no. 3, pp. 264–270, July 1993.    [14] C. D. French, P. P. Acarnley, and A. G. Jack, “Real-time current estimation in brushless DC drives using a single DC-link current sensor,” EPE Conf Rec., 1993, pp. 445–450.    [15] J. F. Moynihan, S. Bolognani, R. C. Kavanagh, M. G. Egan, and J. M. D. Murphy, “Single sensor current control of AC servo drives using digital signal processors,” EPE Conf Rec., 1993, pp. 415–421.    [16] J. Zhang and M. Schroff, “Current control of three-phase brushless DC drives with DC-link current measurement,” Power Conv. Intell. Motion (PCIM) Conf Rec., pp. 141–148, June 1997.    [17] F. Blaabjerg, J. K. Pedersen, U. Jaeger, and P. Thoegersen, “Single current sensor technique in the DC link of three-phase PWM-VS inverters: A review and a novel solution,” IEEE Trans. Ind. Applicat., vol. 33, no. 5, pp. 1241–1253, September/October 1997.    [18] H. Tan and S. L. Ho, “A novel single current sensor technique suitable for BLDCM drives,” in Proc. IEEE-PEDS Conf, 1999, pp. 133–138.    [19] L. Ying and N. Ertugrul, “A novel estimation of phase currents from DC link for permanent magnet AC motors,” Conf Rec., pp. 606–612, 2001.    [20] T. M. Wolbank and P. Macheiner, “An improved observer-based current controller for inverter fed AC machines with single DC-link current measurement,” in Proc. IEEE-PESC Conf, 2002, pp. 1003–1008.    [21] Z. Yu, “Phase current sensor using inverter leg shunt resistor,” US patent, U.S. Pat. No. 6,529,393, Texas Instruments Incorporated, Mar. 4 2003.
Based on the concept of SVM, an inverter feeding the motor has only eight possible switching states represented with two zero-states and six active-states. During the six active states, only one of three phase currents flows through the DC link. At two zero-states, however, the phase currents circulate in the inverter bridge though the diode, not passing through the DC link. Under PWM current control mode, there are two possible active-states in every modulation period. So, two-phase currents can be derived from the DC link current. However, under certain operating conditions of the PWM control, either two active-states may last very short period of time. Therefore, due to the finite switching time of the power devices, the dead time, and the delays in the electronic circuits, actual phase current may not be visible on the dc link measurement.
FIG. 5 is a block diagram of a conventional six-step motor driver in which the motor driver includes A-phase, B-phase, and C-phase upper side drive transistors 101, 103, and 105, U-phase, V-phase, and W-phase lower side drive transistors 102, 104, and 106, diodes 101D, 102D, 103D, 104D, 105D, and 106D, a Hall sensor circuit 201, a conventional six-step control circuit 202, a pre-drive circuit 203, and a current detection resistor 204. A motor includes a A-phase coil 301, a B-phase coil 302, and a C-phase coil 303.
In this embodiment N-type metal oxide semiconductor (NMOS) transistors are used as the drive transistors 101–106. The anode end and cathode end of the diode 101D are connected to the source terminal and drain terminal of the drive transistor 101 respectively. Likewise, the anode end and cathode end of the diode 102D–106D are connected to the source terminal and drain terminal of the drive transistors 102–106 respectively in the same manner. The drains terminal of the drive transistors 101, 103, and 105 are connected to the power supply Vcc, and the source terminals of the drive transistors 102, 104, and 106 are connected to one end of the current detection resistor 205. The other end of the current detection resistor 204 is grounded. The arm of the drive transistors 101–102 and the diodes 101D–102D operate as a A-phase output circuit, the arm of the drive transistors 103–104 and the diodes 103D–104D operate as a B-phase output circuit, and the arm of the drive transistors 105–106 and the diodes 105D–106D operate as a C-phase output circuit. The common node of the source terminal of the transistor 101 and the drain terminal of the transistor 102 is connected to one terminal of the A-phase coil 301. Likewise, the common node of the source terminal of the transistor 103 and the drain terminal of the transistor 104 is connected to one terminal of the B-phase coil 302, and the common node of the source terminal of the transistor 105 and the drain terminal of the transistor 106 is connected to one terminal of the C-phase coil 303. The other terminals of the A-phase coil 301, the B-phase coil 302, and the C-phase coil 303 are connected to one another.
The current flowing from the drive transistors 101–102 toward to the A-phase coil 301 is called a A-phase current IA. Likewise, the current flowing from the drive transistors 103–104 toward to the B-phase coil 302 is called a B-phase current IB, and the current flowing from the drive transistors 105–106 toward to the C-phase coil 303 is called a C-phase current IC. The direction of all the phase currents IA, IB, and IC toward from the drive transistors 101–106 toward to the coils 301–303 is assumed as the positive direction for all the phase currents. The coils 301–303 of the motor 300 are in Y connection. Therefore, the respective phase currents are equal to currents flowing through the corresponding coils.
The Hall sensor circuit 201 includes Hall sensors 201A, 201B, and 201C, which detect the position of a rotor of the motor 300 and output the detection results to the position detection circuit and current command generation circuit 22 as Hall sensors 201A, 201B, and 201C output H1+, H1−, H2+, H2−, H3+, and H3−. The conventional six-step control circuit 202, which receives the Hall sensor outputs H1+, H1−, H2+, H2−, H3+, and H3−, a torque command signal TC, and a feedback current signal Ifb, generates switching control signals S11-S16 to select any of the drive transistors 101–106 to be turned on or off, and sends instructions to the pre-drive circuit 203. The pre-drive circuit 203 outputs signals to the gates of the drive transistors 101–106 according to the outputs of the conventional six-step control circuit 202 in order to control ON/OFF of the drive transistors 101–106.
FIG. 6 is a block diagram of a conventional six-step control circuit in which the six-step control circuit includes differential amplifiers 401A, 401B, and 401C, auto gain control circuits 402A, 402B, and 402C, adders 403A, 403B, and 403C, multipliers 404A, 404B, and 404C, comparators 405A, 405B, 405C, 412A, 412B, and 412C, a low pass filter 406, a peak detection circuit 407, an adder 408, controller 409, a carrier signal generator 410, and a dead time control circuit 411. Differential amplifiers 401A, 401B, and 401C, which receive the Hall sensor outputs H1+, H1−, H2+, H2−, H3+, and H3− respectively, determines the position signals Ha, Hb, and Hc based on the Hall sensor outputs H1+, H1−, H2+, H2−, H3+, and H3−, and outputs the position signals Ha, Hb, and Hc to the auto gain control circuits 402A, 402B, and 402C. The auto gain control circuits 402A, 402B, and 402C adjust the magnitudes of the position signals Ha, Hb, and Hc and then generate signals H11, H21, and H31. The adders, which receive signals H11, H21, and H31, generate signals H13, H23, and H33 to the comparators 412A, 412B, and 412C respectively. The low pass filter 406, which receive a current feedback signal Ifb, outputs a signal to the peak detection circuit 407. The adder 408, which receives the torque command signal TC and the detection result generated by the peak detection circuit 407, outputs the error signal to the controller 409. The multipliers 404A, 404B, and 404C, which receive the output signal of the controller 409 and the output signals of the comparators 412A, 412B, and 412C respectively, output the results to the comparators 405A, 405B, and 405C respectively. The dead time control circuit 411 determines the switching control signals S11–S16 based on the outputs of the comparators 405A, 405B, and 405C.
FIG. 6 shows the control block diagram of the conventional current control architecture for spindle motors. The fundamental of this control scheme is similar to the open loop voltage/frequency control. The amplitudes and phases of the voltages are controlled separately. There are several limitations of this control scheme. Since the dc-link current depends on the PWM signals, a discontinuous current is measured as current feedback as shown in FIG. 7. After detecting the peak value of the dc-link current, a continuous current feedback can be generated as shown in FIG. 7. However, the generated current feedback contains large ripples, which may cause poor current control performance, even at steady-state operations. Besides, the control parameters of the current controller 409 shown in FIG. 6 are required to be tuned for improving the control performance when applying to different motors. FIG. 8 shows the simulation results of current control performance with the conventional six-step control architecture.
FIG. 9 shows the control block diagram of the modified six-step control circuit. Three comparators 412A, 412B, and 412C are omitted from FIG. 6. FIG. 10 shows the simulation waveforms of current control performance with the modified six-step control architecture. Since only the amplitude of the maximum phase current is controlled, the controlled phase current is similar to a trapezoidal waveform as shown in FIG. 10. Besides, because of the non-sinusoidal phase currents, the generated torque contains a torque ripple, which may cause the motor oscillation and may degrade the efficiency.
The conventional approach, either the block modulation or the sinusoidal PWM, suffers from a problem that only the amplitude of the maximum current can be controlled. Therefore, the shape of the phase current cannot be controlled. In U.S. Pat. No. 6,674,258, Matsushita Electric Industrial Co. has proposed a current control architecture which can control two phase currents within one PWM switching period. FIG. 11 shows the overall control block diagram of the Matsushita's approach. For simplicity, three trapezoidal current commands are generated as shown in FIG. 12.
Take the time interval TU1 in FIG. 12 as an example to explain the fundamental of this control approach. During this time interval, the terminal voltage for the phase a is forced to Vcc as shown in FIG. 13(a), and the phase current ia is required to be controlled to the torque current command TI. Since only one phase current can be sensed from the dc-link current, the other two terminal voltages are switched to ground for sensing the phase current ia at the beginning of one PWM switching period as also shown in FIG. 13(a). As ia reaches the torque current command, the lower switch of phase b is turned off by the control signal F1, and the phase current ib flows through the diode 3D of the upper switch as shown in FIG. 13(b). After F1 switched off, the negative of the phase current ic can be sensed from the dc-link current, and is controlled to follow the ramp current command TP as shown in FIG. 14(a)–14(b). As the negative ic reaches the ramp current command TP, the lower switch of phase 3 is turned off by the control signal F2, and the phase current ic flows through the diode SD of the upper switch as shown in FIG. 13(c). In theory, this approach can not only control the amplitude of the maximum phase current, but also can control the shape of one of the other two phase currents during one PWM switching period. FIG. 15 shows the simulation results of the Matsushita's approach. From this figure, the generated torque contains a large torque ripple because of the non-sinusoidal current waveforms. It should be noted that the controlled phase currents are not the desired ideal trapezoidal waveforms as shown in FIG. 12. The reasons will be explained in the following paragraph.
In practice, this approach has a fundamental problem for controlling two phase currents within one PWM switching period. Again, take the time interval TU1 in FIG. 12 as an example. In the beginning of the PWM switching period, phase current ia is controlled towards to the torque current command TI. However, in the meanwhile, the negative of the phase current ic also increases as shown in FIG. 16. When the phase current ia reaches the command, the phase current ic may already exceed the ramp current command as indicated in FIG. 16. Hence the shape of the phase current ic can not be controlled until the ramp current command exceeds the negative of the phase current ic. FIG. 17(a) indicates that even the current commands are three sinusoidal waveforms, this fundamental problem may still occur. Another observation can be made from FIG. 17(b) that if the controllable current shape is ib instead of ic the phase current ib can be controlled until the current command is lower than the negative of the phase current ib. Therefore, one reasonable solution for this fundamental problem is to control ia and ib in the first half of TU1 and to control ia and ic in the second half of TU1. Mathematical analyses will be given to explain the fundamental problem of this approach in the following section.
From FIG. 13(a), three phase voltage equations can be derived as follows:
                              v                      a            ⁢                                                  ⁢            n                          =                                            v              a                        -                          v              n                                =                                                    V                                  c                  ⁢                                                                          ⁢                  c                                            -                                                1                  3                                ⁢                                  (                                                            V                                              c                        ⁢                                                                                                  ⁢                        c                                                              +                    0                    +                    0                                    )                                                      =                                                            i                  a                                ⁢                R                            +                              L                ⁢                                                      ⅆ                                          i                      a                                                                            ⅆ                    t                                                              +                              e                a                                                                        (        2        )                                          v                      b            ⁢                                                  ⁢            n                          =                                            v              b                        -                          v              n                                =                                    0              -                                                1                  3                                ⁢                                  (                                                            V                                              c                        ⁢                                                                                                  ⁢                        c                                                              +                    0                    +                    0                                    )                                                      =                                                            i                  b                                ⁢                R                            +                              L                ⁢                                                      ⅆ                                          i                      b                                                                            ⅆ                    t                                                              +                              e                b                                                                        (        3        )                                          v                      c            ⁢                                                  ⁢            n                          =                                            v              c                        -                          v              n                                =                                    0              -                                                1                  3                                ⁢                                  (                                                            V                                              c                        ⁢                                                                                                  ⁢                        c                                                              +                    0                    +                    0                                    )                                                      =                                                            i                  c                                ⁢                R                            +                              L                ⁢                                                      ⅆ                                          i                      c                                                                            ⅆ                    t                                                              +                              e                c                                                                        (        4        )            where van, vbn, vcn are three phase voltages, va, vb, vc are three terminal voltages, Vcc is the dc-link supply voltage, ia, ib, ic are three phase currents, ea, eb, ec are three back-emf voltages, R and L are the stator resistance and inductance. From the above equations, the variance of the phase currents can be estimated as
                              Δ          ⁢                                          ⁢                      i                          a              ⁢                                                          ⁢              1                                      =                              1            L                    ⁢                      (                                                            2                  3                                ⁢                                  V                                      c                    ⁢                                                                                  ⁢                    c                                                              -                              e                a                            -                                                i                  a                                ⁢                R                                      )                                              (        5        )                                          Δ          ⁢                                          ⁢                      i                          b              ⁢                                                          ⁢              1                                      =                              1            L                    ⁢                      (                                                            -                                      1                    3                                                  ⁢                                  V                                      c                    ⁢                                                                                  ⁢                    c                                                              -                              e                b                            -                                                i                  b                                ⁢                R                                      )                                              (        6        )                                          Δ          ⁢                                          ⁢                      i                          c              ⁢                                                          ⁢              1                                      =                              1            L                    ⁢                      (                                                            -                                      1                    3                                                  ⁢                                  V                                      c                    ⁢                                                                                  ⁢                    c                                                              -                              e                c                            -                                                i                  c                                ⁢                R                                      )                                              (        7        )            Similar analyses can be done for FIG. 13(b) as
                              Δ          ⁢                                          ⁢                      i                          a              ⁢                                                          ⁢              2                                      =                              1            L                    ⁢                      (                                                            1                  3                                ⁢                                  V                                      c                    ⁢                                                                                  ⁢                    c                                                              -                              e                a                            -                                                i                  a                                ⁢                R                                      )                                              (        8        )                                          Δ          ⁢                                          ⁢                      i                          b              ⁢                                                          ⁢              2                                      =                              1            L                    ⁢                      (                                                            1                  3                                ⁢                                  V                                      c                    ⁢                                                                                  ⁢                    c                                                              -                              e                b                            -                                                i                  b                                ⁢                R                                      )                                              (        9        )                                          Δ          ⁢                                          ⁢                      i                          c              ⁢                                                          ⁢              2                                      =                              1            L                    ⁢                      (                                                            -                                      2                    3                                                  ⁢                                  V                                      c                    ⁢                                                                                  ⁢                    c                                                              -                              e                c                            -                                                i                  c                                ⁢                R                                      )                                              (        10        )            and for FIG. 13(c) as
                              Δ          ⁢                                          ⁢                      i                          a              ⁢                                                          ⁢              0                                      =                              1            L                    ⁢                      (                                          -                                  e                  a                                            -                                                i                  a                                ⁢                R                                      )                                              (        11        )                                          Δ          ⁢                                          ⁢                      i                          b              ⁢                                                          ⁢              0                                      =                              1            L                    ⁢                      (                                          -                                  e                  b                                            -                                                i                  b                                ⁢                R                                      )                                              (        12        )                                          Δ          ⁢                                          ⁢                      i                          c              ⁢                                                          ⁢              0                                      =                              1            L                    ⁢                      (                                          -                                  e                  c                                            -                                                i                  c                                ⁢                R                                      )                                              (        13        )            Define the time interval for FIG. 13(a)–(c) as Δtn1, Δtn2, and Δtn3, where n denotes n-th switching period within the time interval TU1. The phase current ic at the k-th switching instant can be derived as follows:
                              i                      c            ⁢                                                  ⁢            k                          =                              ∑                          n              =              1                        k                    ⁢                                          ⁢                      (                                          Δ                ⁢                                                                  ⁢                                  i                                      c                    ⁢                                                                                  ⁢                    n                    ⁢                                                                                  ⁢                    1                                                  ⁢                Δ                ⁢                                                                  ⁢                                  t                                      n                    ⁢                                                                                  ⁢                    1                                                              +                              Δ                ⁢                                                                  ⁢                                  i                                      c                    ⁢                                                                                  ⁢                    n                    ⁢                                                                                  ⁢                    2                                                  ⁢                Δ                ⁢                                                                  ⁢                                  t                                      n                    ⁢                                                                                  ⁢                    2                                                              +                              Δ                ⁢                                                                  ⁢                                  i                                      c                    ⁢                                                                                  ⁢                    n                    ⁢                                                                                  ⁢                    0                                                  ⁢                Δ                ⁢                                                                  ⁢                                  t                                      n                    ⁢                                                                                  ⁢                    0                                                                        )                                              (        14        )            From (7), (10), and (13), (14) can be derived as
                              i                      c            ⁢                                                  ⁢            k                          =                                            -              1                        L                    ⁢                                    ∑                              n                =                1                            k                        ⁢                                                  ⁢                          [                                                                    1                    3                                    ⁢                                                            V                                              c                        ⁢                                                                                                  ⁢                        c                                                              ⁡                                          (                                                                        Δ                          ⁢                                                                                                          ⁢                                                      t                                                          n                              ⁢                                                                                                                          ⁢                              1                                                                                                      +                                                  2                          ⁢                          Δ                          ⁢                                                                                                          ⁢                                                      t                                                          n                              ⁢                                                                                                                          ⁢                              2                                                                                                                          )                                                                      +                                                      e                                                                  c                        ⁢                                                                                                  ⁢                        n                                            ⁢                                                                                                                            ⁢                  Δ                  ⁢                                                                          ⁢                                      T                                          s                      ⁢                                                                                          ⁢                      w                                                                      +                                                      i                                                                  c                        ⁢                                                                                                  ⁢                        n                                            ⁢                                                                                                                            ⁢                  R                  ⁢                                                                          ⁢                  Δ                  ⁢                                                                          ⁢                                      T                                          s                      ⁢                                                                                          ⁢                      w                                                                                  ]                                                          (        15        )            where ΔTsw denotes the switching period, which is also the summation of Δtn1, Δtn2, and Δtn0. With the trapezoidal current waveforms as shown in FIG. 12, the phase current command ic* at the k-th switching instant can be derived as follows:
                              i                      c            ⁢                                                  ⁢            k                    *                =                              -                                          ∑                                  n                  =                  1                                k                            ⁢                                                          ⁢                              Δ                ⁢                                                                  ⁢                                  i                  c                  *                                ⁢                Δ                ⁢                                                                  ⁢                                  T                                      s                    ⁢                                                                                  ⁢                    w                                                                                =                                    -                                                P                  ⁢                                                                          ⁢                                      ω                    0                                                  20                                      ⁢                          i              *                        ⁢            Δ            ⁢                                                  ⁢                          T                              s                ⁢                                                                  ⁢                w                                                                        (        16        )            where P denotes the poles of the spindle motor, and ω0 denotes the rotating speed at the first switching instant, and i* denotes the amplitude of the current command, respectively. As discussed before, if we want to control the phase current ic to follow the current command ic* within the time interval of TU1, then we haveick≧ick*  (17)By substitute (15) and (16) into (17), the condition of (17) can be rewritten as
                                          ∑                          n              =              1                        k                    ⁢                                          ⁢                      [                                                            1                  3                                ⁢                                                      V                                          c                      ⁢                                                                                          ⁢                      c                                                        ⁡                                      (                                          1                      +                                                                        Δ                          ⁢                                                                                                          ⁢                                                      t                                                          n                              ⁢                                                                                                                          ⁢                              2                                                                                                                                Δ                          ⁢                                                                                                          ⁢                                                      T                                                          s                              ⁢                                                                                                                          ⁢                              w                                                                                                                          -                                                                        Δ                          ⁢                                                                                                          ⁢                                                      t                                                          n                              ⁢                                                                                                                          ⁢                              0                                                                                                                                Δ                          ⁢                                                                                                          ⁢                                                      T                                                          s                              ⁢                                                                                                                          ⁢                              w                                                                                                                                            )                                                              +                              (                                                      e                                                                  c                        ⁢                                                                                                  ⁢                        n                                            ⁢                                                                                                                            +                                                            i                                                                        c                          ⁢                                                                                                          ⁢                          n                                                ⁢                                                                                                                                        ⁢                    R                                                  )                                      ]                          ≤                                            P              ⁢                                                          ⁢                              ω                0                            ⁢              L                        20                    ⁢                      i            *                                              (        18        )            If the equation (18) stands at k-th switching instant, the phase current ic can be controlled to follow the current command ic* after k-th switching instant within the time interval of TU1. Hence the equation (18) is the condition for determining whether the shape of the phase current ic is controllable or not. Some observations can be made from (18) as follows. Within the time interval of TU1, the first term in the left-hand side of (18) is positive and the second term is negative, that is:
                    0        <                              1            3                    ⁢                                    V                              c                ⁢                                                                  ⁢                c                                      ⁡                          (                              1                +                                                      Δ                    ⁢                                                                                  ⁢                                          t                                              n                        ⁢                                                                                                  ⁢                        2                                                                                                  Δ                    ⁢                                                                                  ⁢                                          T                                              s                        ⁢                                                                                                  ⁢                        w                                                                                            -                                                      Δ                    ⁢                                                                                  ⁢                                          t                                              n                        ⁢                                                                                                  ⁢                        0                                                                                                  Δ                    ⁢                                                                                  ⁢                                          T                                              s                        ⁢                                                                                                  ⁢                        w                                                                                                        )                                      <                              1            3                    ⁢                      V                          c              ⁢                                                          ⁢              c                                                          (        19        )            ecn+icnR<0  (20)
The right-hand side of (18) is directly proportional to the rotating speed ω0 and the amplitude i* of the phase current. Therefore, the equation (18) is much easier to be satisfied at higher speeds than at lower speeds as shown in FIG. 18(a)–18(b). Assume that the back-EMF voltage and the phase current are both in sinusoidal shapes, and can be derived for the time interval TU1 as follows
                              e                      c            ⁢                                                  ⁢            n                          =                                            -                              K                E                                      ⁢                          ω              n                        ⁢                          sin              ⁡                              (                                                      60                    ⁢                    n                    ⁢                                                                                  ⁢                    Δ                    ⁢                                                                                  ⁢                                          T                                              s                        ⁢                                                                                                  ⁢                        w                                                                                                  20                                          P                      ⁢                                                                                          ⁢                                              ω                        0                                                                                            )                                              =                                    -                              K                E                                      ⁢                          ω              n                        ⁢                          sin              ⁡                              (                                  3                  ⁢                  P                  ⁢                                                                          ⁢                                      ω                    0                                    ⁢                  n                  ⁢                                                                          ⁢                  Δ                  ⁢                                                                          ⁢                                      T                                          s                      ⁢                                                                                          ⁢                      w                                                                      )                                                                        (        21        )            icn=−i*sin(3Pω0nΔTsw)  (22)
At low-speeds, the right-hand side of (18) is approximately zero. Therefore for the negative summation in the left-hand side of (18), we have
                                          (                                                            K                  E                                ⁢                                  ω                  n                                            +                              i                *                                      )                    ⁢                      sin            ⁡                          (                              3                ⁢                P                ⁢                                                                  ⁢                                  ω                  0                                ⁢                n                ⁢                                                                  ⁢                Δ                ⁢                                                                  ⁢                                  T                                      s                    ⁢                                                                                  ⁢                    w                                                              )                                      ≥                              1            3                    ⁢                                    V                              c                ⁢                                                                  ⁢                c                                      ⁡                          (                              1                +                                                      Δ                    ⁢                                                                                  ⁢                                          t                                              n                        ⁢                                                                                                  ⁢                        2                                                                                                  Δ                    ⁢                                                                                  ⁢                                          T                                              s                        ⁢                                                                                                  ⁢                        w                                                                                            -                                                      Δ                    ⁢                                                                                  ⁢                                          t                                              n                        ⁢                                                                                                  ⁢                        0                                                                                                  Δ                    ⁢                                                                                  ⁢                                          T                                              s                        ⁢                                                                                                  ⁢                        w                                                                                                        )                                                          (        23        )            
From the condition of (23), one concluding mark can be made that the equation (23) can be only satisfied with a sufficiently large n at low-speed operations, that is, controlling the shape of the phase current ic is not possible within the whole time interval of TU1 at low-speed operations. This phenomenon may induce torque ripple to affect the overall control performance.
From the above analyses, the concept of the Matsushita's approach has several advantages. First, not only the amplitudes, but also the shapes of the phase currents are possible to be controlled to reduce the torque ripple. Second, no control parameter is required to be tuned. Third, only two phases are required to be switched at any instant, hence the switching losses of the power transistors can be reduced. However, the Matsushita's approach consists of a fundamental problem for controlling the current shapes. Therefore, new control architecture is proposed to reserve the advantages and improve the weakness of the Matsushita's approach.